The equation of the smallest circle passing through the intersection of the line x+y=1  and the circle x2+y2=9  is:

S=x2+y2−9=0 , L=x+y−1=0 .p=1/2<3=r    ∴    Line intersects the circle S .Equation of circle passing through the points of intersections of S=0  and L=0  is  (x2+y2−9)+λ(x+y−1)=0 .This circle is smallest if chord x+y−1=0 becomes diameter i.e., centre (−λ2,−λ2) lies on it.∴     −λ2−λ2−1=0⇒λ=−1 .∴    required circle is x2+y2−x−y−8=0 .